This review summarizes recent progress in understanding the equilibration and thermalization of closed quantum many-body systems out of equilibrium, focusing on quenches, ramps, and periodic driving. It addresses key topics such as the eigenstate thermalization hypothesis (ETH), typicality, transport, many-body localization (MBL), universality near phase transitions, and prospects for quantum simulations. The article discusses how closed quantum systems, initially in a non-equilibrium state, eventually equilibrate and thermalize, and explores the underlying mechanisms, including the role of locality, integrability, and the breakdown of thermalization in MBL systems. It also highlights the importance of quantum simulations in probing these phenomena, with a focus on cold atoms and trapped ions. The review emphasizes the challenges in understanding the time scales of equilibration and the role of quantum information theory and thermodynamics in this context. It concludes with perspectives on the future of research, including the potential of quantum simulators to outperform classical computers in simulating out-of-equilibrium dynamics.This review summarizes recent progress in understanding the equilibration and thermalization of closed quantum many-body systems out of equilibrium, focusing on quenches, ramps, and periodic driving. It addresses key topics such as the eigenstate thermalization hypothesis (ETH), typicality, transport, many-body localization (MBL), universality near phase transitions, and prospects for quantum simulations. The article discusses how closed quantum systems, initially in a non-equilibrium state, eventually equilibrate and thermalize, and explores the underlying mechanisms, including the role of locality, integrability, and the breakdown of thermalization in MBL systems. It also highlights the importance of quantum simulations in probing these phenomena, with a focus on cold atoms and trapped ions. The review emphasizes the challenges in understanding the time scales of equilibration and the role of quantum information theory and thermodynamics in this context. It concludes with perspectives on the future of research, including the potential of quantum simulators to outperform classical computers in simulating out-of-equilibrium dynamics.